Expression For The Excess Pressure Due To Water Hammer
When a valve with a volumetric flow rate Q is closed, an excess pressure δP is created upstream of the valve, whose value is given by the Joukowsky equation:
In this expression:
- overpressurization δP is expressed in Pa;
- Q is the volumetric flow in m3/s;
- Zh is the hydraulic impedance, expressed in kg/m4/s.
The hydraulic impedance Zh of the pipeline determines the magnitude of the water hammer pulse. It is itself defined by:
with:
- ρ the density of the liquid, expressed in kg/m3;
- A cross sectional area of the pipe, m2;
- Beff effective modulus of compressibility of the liquid in the pipe, expressed in Pa.
The latter follows from a series of hydraulic concepts:
- compressibility of the liquid, defined by its adiabatic compressibility modulus Bl, resulting from the equation of state of the liquid generally available from thermodynamic tables;
- the elasticity of the walls of the pipe, which defines a modulus of equivalent compressibility Beq. In the case of a pipe of circular cross section whose thickness e is small compared to the diameter D, the equivalent modulus of compressibility is given by the following formula: ; in which E is the Young's modulus (in Pa) of the material of the pipe;
- possibly compressibility Bg of gas dissolved in the liquid, defined by:
- γ being the ratio of specific heats of the gas
- α the rate of ventilation (the volume fraction of undissolved gas)
- and P the pressure (in Pa).
Thus, the effective compressibility modulus is:
As a result, we see that we can reduce the water hammer by:
- increasing the pipe diameter at constant flow, which reduces the inertia of the liquid column;
- choosing to use a material with a reduced Young's modulus;
- introducing a device that increases the flexibility of the entire hydraulic system, such as a hydraulic accumulator;
- where possible, increasing the percentage of undissolved air in the liquid.
Read more about this topic: Water Hammer
Famous quotes containing the words expression, excess, pressure, due, water and/or hammer:
“You have to begin to lose your memory, if only in bits and pieces, to realize that memory is what makes our lives. Life without memory is no life at all, just as an intelligence without the possibility of expression is not really an intelligence. Our memory is our coherence, our reason, our feeling, even our action. Without it, we are nothing.”
—Luis Buñuel (1900–1983)
“Genius has infused itself into nature. It indicates itself by a small excess of good, a small balance in brute facts always favorable to the side of reason.”
—Ralph Waldo Emerson (1803–1882)
“He who is of a calm and happy nature will hardly feel the pressure of age, but to him who is of an opposite disposition youth and age are equally a burden.”
—Plato (c. 427–347 B.C.)
“Praise is a more ingenious, concealed, and subtle kind of flattery, that satisfies both the giver and the receiver, though by very different ways. The one accepts it as a reward due to his merit; the other gives it that he may be looked upon as a just and discerning person.”
—François, Duc De La Rochefoucauld (1613–1680)
“A single monk carries his own bucket of water; two monks carry their bucket of water together; but when there are three monks, there is no water at all.”
—Chinese proverb.
“I suppose that one of the psychological principles of advertising is to so hammer the name of your product into the mind of the timid buyer that when he is confronted with a brusk demand for an order he can’t think of anything else to say, whether he wants it or not.”
—Robert Benchley (1889–1945)